Answer to Question #190481 in Differential Equations for Cecilia

Question

Complete solution of d^2y/dx^2 + 4y = cos 4x

Answer 1

Let us complete solution of differential equation $\frac{d^2y}{dx^2} + 4y = \cos 4x.$

Firstly, let us solve the characteristic equation of homogeneous differential equation:

$k^2+4=0$

$k_1=2i,\ k_2=-2i.$

Therefore, the general solution of homogeneous equation is

$y=C_1\cos 2x + C_2\sin 2x.$

Let us find the partial solution of non-homogeneous equation:

$y_p=A\cos 4x+B\sin 4x$

$y_p'=-4A\sin 4x+4B\cos 4x$

$y_p''=-16A\cos 4x-16B\sin 4x$

$-16A\cos 4x-16B\sin 4x+4A\cos 4x +4B\sin 4x=\cos 4x$

$-12A\cos 4x-12B\sin 4x=\cos 4x$

$-12A=1, -12B=0$

$A=-\frac{1}{12}, \ B=0.$

Consequently the general solution of the differential equation $\frac{d^2y}{dx^2} + 4y = \cos 4x$ is

$y=C_1\cos 2x + C_2\sin 2x-\frac{1}{12}\cos 4x.$